Hostname: page-component-848d4c4894-xm8r8 Total loading time: 0 Render date: 2024-07-01T02:24:43.907Z Has data issue: false hasContentIssue false
  • English
  • Français

Symmetric groupoids, free categories and E*-unitary inverse monoids

Published online by Cambridge University Press:  14 November 2011

Jonathan Leech
Jonathan Leech
Affiliation:
Department of Mathematics, Westmont College, 955 La Paz Road, Santa Barbara, CA 93108-1099, USA ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Symmetric groupoids which classify the monomorphism contexts of objects in arbitrary categories are studied, along with their connections to symmetric inverse monoids and symmetric inverse algebras. Particular attention is given to symmetric groupoids of objects in free categories and to the inverse algebras induced from them by 0-closure. These graph algebras generalize both the class of polycyclic semigroups and the class of combinatorial ω-semigroups with adjoined zeros. Since all such algebras are E*-unitary, an analogue of McAlister's theory of.E-unitary inverse monoids is developed for a special class of E*-unitary inverse monoids and then illustrated on the class of graph algebras.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 5 , 1999 , pp. 959 - 984
Copyright
Copyright © Royal Society of Edinburgh 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Ehiesmann, C. H.. Categories inductives et pseudogroupes. Ann. Inst. Fourier Grenoble 10 (1960), 307–322.CrossRefGoogle Scholar
2FitzGerald, D. G.. Normal bands and their inverse semigroups of bicongruences. J. Algebra 153 (1996), 502–526.Google Scholar
3FitzGerald, D. G. and Leech, J.. Dual symmetric inverse monoids and representation theory. J. Australian Math. Soc. A 64 (1998), 345–367.CrossRefGoogle Scholar
4Gomes, G. M. S. and Howie, J. M.. A P-theorem for inverse semigroups with zero. Portugaliae Mathematica 53 (1996), 257–278.Google Scholar
5Grillet, P. A.. Semigroups, an introduction to the structure theory (New York: Marcel Dekker, 1995).Google Scholar
6Higgins, P. M.. Techniques of semigroup theory (Oxford University Press, 1992).CrossRefGoogle Scholar
7Howie, J. M.. Fundamentals of semigroup theory (Oxford University Press, 1995).CrossRefGoogle Scholar
8Kochin, B. P.. The structure of inverse ideal-simple semigroups. Vestnik Leningrad Univ. 23 (1968). 41–50 (Russian).Google Scholar
9Lawson, M. V.. The geometric theory of inverse semigroups. I. J. Pure Appl. Algebra 67 (1990), 151–177.CrossRefGoogle Scholar
10Lawson, M. V.. Constructing inverse semigroups from category actions. J. Pure Appl. Algebra 137 (1999), 57–101.CrossRefGoogle Scholar
11Leech, J.. The D-category of a monoid. Semigroup Forum 34 (1986), 89–116.CrossRefGoogle Scholar
12Leech, J.. Constructing inverse monoids from small categories. Semigroup Forum 36 (1987), 89–116.CrossRefGoogle Scholar
13Leech, J.. Inverse monoids with a natural semi-lattice ordering. Proc. Lond. Math. Soc. 70 (1995), 146–182.CrossRefGoogle Scholar